shopping

I'm considering buying a bike to finish a project I started with a backpack and good hiking shoes.
So far, my daughter and I have completed 300 miles of the Arizona trail on foot. Five hundred to go, including the Grand Canyon, (and I'm not carrying a bike down and back up).
For a number of reasons, I've begun thinking about finishing the trail between two wheels.
Some of the riding will be on 2-track roads, some on trail with varying tread, and maybe some on hard-top.
I'd appreciate some guidance on the type of bike to look for, without spending 4 figures.
Many thanks.